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Unformatted text preview: Lecture 26 Daniel Apon 1 From IP=PSPACE to NP=PCP( log , 1) : NEXP has multi-prover interactive protocols If you’ve read the notes on the history of the PCP theorem referenced in Lecture 19 , you will already be familiar with the excitement that surrounded discoveries such as IP=PSPACE and NP=PCP( log , 1) . Taken alone, however, these two theorems might seem at most cursorily related to one another. The earlier discovery of IP=PSPACE deals with a much stronger class, assuming the polynomial hierarchy doesn’t collapse, than the NP version of the PCP theorem. The former characterizes the verification power of a Turing machine with an adaptive, all- powerful prover, while the latter studies an efficient, probabilistic notion of constant-query proof checking with applications to inapproximability of NP-complete problems, and so on. But in fact, a series of natural generalizations lead from one to the other. An immediate step after IP=PSPACE is to ask what happens in the case of multiple , all-powerful provers interacting with a single PPT verifier. In particular, for what power of computational prob- lems can a multi-prover interactive proof system correctly verify candidate-solutions with high probability? As it turns out, we can (and will, in this lecture) show that MIP=NEXP , where MIP is the class of languages who have a multi-prover interactive proof and NEXP is the class of languages that can be decided in nondeterministic exponential time. The techniques involved are similar to that in the proof IP=PSPACE ; however, the fact that we want to reliably verify exponential-sized objects using a PPT machine requires new ideas. We will show that the fact that the verifier interacts with multiple provers (who can speak with the verifier but may not speak with one another during the course of the protocol) gives us the desired power, because the verifier has access to an additional consistency test – namely, re-query new provers on a random subset of the verifier’s queries made during the course of some protocol. Just beyond the scope of this lecture, the ideas used in the proof of MIP=NEXP can be used to prove NEXP=PCP( poly, poly ) . A series of attempts to “scale down” this result to NP eventually produced the PCP theorem of Arora, et al. (See  for more on this.) In the sequel, we prove MIP=NEXP following the original paper by Babai, Fortnow, and Lund  and Lund’s PhD thesis . 2 Preliminaries Formally, we define the class of decision problems MIP as follows. Let the provers P 1 ,...,P k be computationally unbounded Turing machines. Let the verifier V be a PPT machine. We allow each P i to communicate with V and vice versa during the course of the protocol, but 1 we do not allow the P i to communicate with one another. Then we say L ∈ MIP if x ∈ L = ⇒ Pr[ P 1 ,...,P k convince V to accept x] = 1 x 6∈ L = ⇒∀ P 1 ,...,P k , Pr[ P 1 ,...,P k convince V to accept x] ≤ 1 / 2 where the P notation denotes a dishonest prover and where the constants are arbitrary.notation denotes a dishonest prover and where the constants are arbitrary....
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- Fall '08
- Computational complexity theory, NEXP